The White Bank

"The White Bank ... was located on the town square in Lancaster ... The White Bank was forced to cease operations in 1930's when all private banks were closed due to the Great Depression. The Bank reopened as The First National Bank of Lancaster."
[https://www.lancaster-tx.com/545/About-the-Library]

This suggests a merger with the already existing First National Bank of Lancaster.

R. P. Henry $ Sons Bank
Est. 1889

"February 27, 1934. Clyde Barrow and fellow gang member Raymond Hamilton rob the R.P. Henry & Sons Bank, in Lancaster's Town Square, taking over $4000. Olin Worley, a customer in the bank, and Laurence L. Henry, one of R.P. Henry's sons and the cashier, were held at gunpoint while the bank was looted."
[Posted on November 26, 2013 by LHSAdmin]

Although these two banks were on the Town Square, the First National Bank apparently was not.
There is no image of it.

The First National Bank of Lancaster, chartered 11423, opened in August, 1919 and printed currency through to the end of the National Bank period. It printed 1902 Plain Backs and 1929 Types 1 & 2. Seventeen remain in the census: eight Plain Backs, six Type 1's, and three Type 2's.

This 1929 Type 2 sold at a Heritage auction in April, 2014, for $1,527.